Newton's law of cooling - formula for constant k

[FEAnalyst]

Hi,

recently I got interested with practical applications of Newton's law of cooling. Its main disadvantage is that one has to measure temperature at some point of time (other than $t=0$) to obtain solution for any other $t$. However I've found an online calculator (https://www.omnicalculator.com/physics/Newtons-law-of-cooling). It features the standard equation from this law: $T=T_{amb}+(T_{0}-T_{amb}) \cdot e^{-k \cdot t}$. However there's also a formula for approximate constant $k$: $$k=\frac{hA}{C}$$ where: $h$ - heat transfer coefficient, $A$ - area of heat exchange, $C$ - heat capacity. If this formula is correct then it makes Newton's law of cooling much more useful since no measurement is needed to obtain approximate results. However my question is - do you know how this formula is derived, what is the reason for such relationship and, what's even more important, where else can I find it (so far I've only seen it on this single website) ? Do you know any books with this equation ? I need some reliable source.

Thanks in advance for your help

 

[anorlunda]

How will you get the value of h without measurements?

 

[Chestermiller]

Check out Transport Phenomena by Bird, Stewart, and Lightfoot

 

[FEAnalyst]

Approximate values of heat transfer coefficient $h$ for various conditions (forced/natural convection) and types of fluid/gas can be easily found in the tables. Of course to get more accurate values one would have to perform some experimental testing but for me these approximate coefficients are enough.

Thanks for this book recommendation. I will try to find it. If you know about any other books with this formula, please let me know.

 

[Chestermiller]

Approximate values of heat transfer coefficient $h$ for various conditions (forced/natural convection) and types of fluid/gas can be easily found in the tables. Of course to get more accurate values one would have to perform some experimental testing but for me these approximate coefficients are enough.

Thanks for this book recommendation. I will try to find it. If you know about any other books with this formula, please let me know.

In many cases, there are dimensionless correlations available of heat transfer coefficient in terms of the Nussult number as a function of Reynolds number and Prantdl number. For natural convection, they are in terms of Grashoff number. So, for the specific system geometry and material properties in your situation, you can get an accurate estimate of the heat transfer coefficient.

See also the Chemical Engineers' Handbook, Mechanical Engineers Handbook, etc.